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          考研数二
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        <h1 id="考研数二"><a href="#考研数二" class="headerlink" title="考研数二"></a>考研数二</h1><h2 id="线代资料篇"><a href="#线代资料篇" class="headerlink" title="线代资料篇"></a>线代资料篇</h2><span id="more"></span>
<h3 id="题型解法"><a href="#题型解法" class="headerlink" title="题型解法"></a>题型解法</h3><h4 id="求线性方程组的解"><a href="#求线性方程组的解" class="headerlink" title="求线性方程组的解"></a>求线性方程组的解</h4><h6 id="齐次方程组的通解"><a href="#齐次方程组的通解" class="headerlink" title="齐次方程组的通解"></a>齐次方程组的通解</h6><p>形式$Ax=0;$</p>
<script type="math/tex; mode=display">
\left\{\begin{aligned}
&有唯一解(零解):&r(A)=n;\\
&有非零解:&r(A)<n;\\
\end{aligned}\right.</script><p>基础解系$\xi_1,\xi_2,\cdot\cdot ,\xi_s\ 的s=n-r(A)$ ;</p>
<p>方法:</p>
<script type="math/tex; mode=display">
\begin{aligned}
&①\ A\stackrel{初等行变换}{\longrightarrow}梯形矩阵B(或最简阶梯形矩阵B)\ ; \\
&②\ 按列找出一个秩为r的子矩阵，则剩余列位置的未知数即设为自由变量(个数S=n-r)\ ;\\
&③\ 按基础解系定义求出\xi_1,\xi_2,\cdot\cdot\cdot,\xi_s并写出通解;
\end{aligned}</script><h6 id="非齐次方程组的通解"><a href="#非齐次方程组的通解" class="headerlink" title="非齐次方程组的通解"></a>非齐次方程组的通解</h6><p>形式$Ax=b$ ;</p>
<script type="math/tex; mode=display">
\begin{cases}
r(A)\neq r(A,b),方程无解\\
r(A)=r(A,b)\begin{cases} 
\ =n:有唯一解\\
\ <n:无穷解
\end{cases}
\end{cases}</script><p>求解步骤</p>
<script type="math/tex; mode=display">
\begin{aligned}&①求Ax=0的通解,k_1\xi_1+k_2\xi_2+\cdot\cdot\cdot+k_{n-r}\xi_{n-r}\\
&②求Ax=b\ 的一个特解\eta\\
&③则通解为k_1\xi_1+k_2\xi_2+\cdot\cdot\cdot+k_{n-r}\xi_{n-r}+\eta\end{aligned}</script><h4 id="求特征值"><a href="#求特征值" class="headerlink" title="求特征值"></a>求特征值</h4><h3 id="归纳总结"><a href="#归纳总结" class="headerlink" title="归纳总结"></a>归纳总结</h3><ul>
<li><p>若矩阵A的行列式$|A|\neq 0$ 向量<strong>线性无关</strong></p>
</li>
<li><p>特征值与特征向量</p>
<ul>
<li><p>​    特征值的性质</p>
<ul>
<li><script type="math/tex; mode=display">
①\sum_{i=1}^{n}\lambda _{i}= \sum _{i=1}^{n}a_{ii}=tr(A);\\
②\prod _ { i = 1 } ^ { n } \lambda_ { i } = | A |</script></li>
</ul>
</li>
<li><p>特征向量的性质</p>
<ul>
<li><strong>(1)k重特征值$\lambda$至多只有k个线性无关的特征向量</strong>（<strong>直接使用，不用证明</strong>）;</li>
<li>(2)若$\xi_1,\xi_2$是A的属于不同特征值$\lambda_1,\lambda_2$的特征向量，<strong>则$\xi_1,\xi_2$线性无关</strong>;</li>
<li>(3)若$\xi_1,\xi_2$是A的属于同一特征值入的特征向量，则$k_1\xi_1+k_2\xi_2$($k_1,k_2$不同时为零的任意常数)仍是A的属于特征值$\lambda$的特征向量;</li>
</ul>
</li>
</ul>
</li>
<li><p>相似矩阵的性质$A\sim B$</p>
<ol>
<li><p>$r(A)=r(B)$;</p>
</li>
<li><p>$|A|=|B|$;</p>
</li>
<li><p>$|\lambda E-A|=|\lambda E-B|$;</p>
</li>
<li><p>$\lambda_{Ai}=\lambda_{Bi}$</p>
</li>
</ol>
</li>
</ul>
<ul>
<li>二次型<ul>
<li>正惯性指数: 将二次型化成规范型或标准型,正项个数为p,负项个数q 。则p为正惯性指数.</li>
</ul>
</li>
</ul>

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